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We have seen that
is the probability density of a
measurement of a particle's displacement yielding the value at time .
Suppose that we made a large number of independent measurements of the
displacement on
an equally large number of identical quantum systems. In general, measurements
made on different systems will yield different results. However, from the
definition of probability, the
mean of all these results is simply

(158) 
Here,
is called the expectation value of .
Similarly the expectation value of any function of is

(159) 
In general, the results of the various different measurements of will be scattered
around the expectation value
. The
degree of scatter is parameterized by the quantity

(160) 
which is known as the variance of . The squareroot of this
quantity, , is called the standard deviation of .
We generally expect the results of measurements of to lie
within a few standard deviations of the expectation value.
For instance, consider the normalized Gaussian wave packet [see Eq. (146)]

(161) 
The expectation value of associated with this wavefunction is

(162) 
Let
. It follows that

(163) 
However, the second integral on the righthand side is zero, by symmetry.
Hence, making use of Eq. (144), we obtain

(164) 
Evidently, the expectation value of for a Gaussian wave packet is
equal to the most likely value of (i.e., the value of which
maximizes ).
The variance of associated with the Gaussian wave packet (161)
is

(165) 
Let
. It follows that

(166) 
However,

(167) 
giving

(168) 
This result is consistent with our earlier interpretation of as a measure of the
spatial extent of the wave packet (see Sect. 3.12).
It follows that we can rewrite the Gaussian wave packet (161) in the convenient form

(169) 
Next: Ehrenfest's Theorem
Up: Fundamentals of Quantum Mechanics
Previous: Normalization of the Wavefunction
Richard Fitzpatrick
20100720